FI-- J-supplemented module s

A Module M is called cofinite JSupplemented Module if for every cofinite submodule L of M, there exists a submodule N of M such that M=L+N with main properties of cof-J-supplemented modules. An R-module M is called fully invariant-J-supplemented if for every fully invariant submodule N of M, there exists a submodule K of M, such that M = N + K with N K K. A condition under which the direct sum of FI-J-supplemented modules is FI-J-supplemented was given. Also, some types of modules that are related to the FI-J-supplemented module were discussed.


Introduction
Throughout this paper , an arbitrary associative ring with identity is denoted by R and all modules are unitary left R-modules . Assume that C and D are submodules of M, a submodule C is called small submodule of M (C , if whenever M = C + D , we have M = D [1]. A submodule C of a module M is called J-small submodule of M if whenever M = C + D , with J( implies M = D , were J(M) denotes the Jacobson radical of M [2]. A submodule C is a supplement of D in M if C is minimal with respect to M = C + D . Equivalently, M = C + D with [3]. A module M is called supplemented module if every submodule of M has a supplement in M [4] . A Submodule C is called J-supplement of D in M if M = C + D and . M is called J-supplemented if every submodule of M has J-supplement in M [2]. A module M is called -supplemented module if every submodule of M has a direct summand supplement in M [5]. A Submodule C is called a -Jacobsonsupplement of D in M (for short -J-supplement ) if M = C + D , and C is a direct summand of M

Ali and Khalid
Iraqi Journal of Science, 2021, Vol. 62, No. 5, pp: 1637-1634 2238 with C ∩ D C . It is called a -J-supplemented if every submodule of M has a -J-supplement in M [6] . A submodule C of a module M is called cofinite submodule of M if is finitely generated . A module M is called cofinitely supplemented if every cofinite submodule of M has supplement submodule [7]. As a generalization of cofinitely supplemented , we define the cofinitely Jsupplemented (for short cof-J-supplemented ) as follows . A module M is called cof-J-supplemented module if for every cofinite submodule C of M , there exists a submodule D of M such that M = C + D and A submodule C of a module M is called a fully invariant submodule if for every [8]. In section 2, we prove some properties of cof-J-supplemented and we show that any factor module of cof-J-supplemented module is cof-J-supplemented and any finite sum of cof-J-supplemented is cof-Jsupplemented . In section 3, we introduce the concept of fully invariant J-supplemented modules ( for short FI-Jsupplemented ) as a generalization of J-supplemented, as follows . The module M is said to be FI-Jsupplemented , if for every fully invariant submodule C of M , there exists a submodule D of M such that M = C + D and . Clearly, the supplemented modules are J-supplemented and the Jsupplemented modules are FI-J-supplemented . As a generalization of a -J-supplemented module , we introduce the concept of fully invariant -J-supplemented modules ( FI--J-supplemented ) . A module M is called FI--J-supplemented , if for every fully invariant submodule C of M , there exists a direct summand D of M such that M = C + D and . Clearly, FI--J-supplemented modules are FI-J-supplemented .

Cofinitely J-supplemented modules
This section is devoted to introduce the cofinitely J-supplemented modules as a generalization of Jsupplemented modules , and illustrate this concept by remarks and properties . (by modular law), but K C C , then [2] . Therefore is a cof-J-supplemented.
The converse in general is not true . For example, Z as Z-module is cof-J-supplemented, but Z is not cof-J-supplemented . Corollary(2.5): The homomorphic image of a cof-J-supplemented module is a cof-J-supplemented module . Proof: Since every homomorphic image is isomorphic to a quotient module . , which is finitely generated , hence is a cofinite submodule of . Since is a cof-J-supplemented , then there exists a submodule H of such that with . We have and since is a cofinite submodule of and is a cof-J-supplemented , then there exists a submodule G of such that and . Then and . Therefore M is a cof-J-supplemented module . by Corollary (2.4). To show that the arbitrary sum of a cof-J-supplemented is cof-J-supplemented , we need the following standard lemma . Lemma . Therefore H + G is J-supplement of N in M .

FI-J-supplemented and FI--J-supplemented modules
In this section , the concept of FI-J-supplemented modules as a generalization of J-supplemented and some properties of this type of modules are given . Also, as a generalization of FI-J-supplemented modules, FI--J-supplemented modules are introduced . Then Therefore is FI-J-supplemented . Proposition (3.5) : Let and U be fully invariant submodules of M , and let be FI-Jsupplemented module . If + U has FI-J-supplement in M, then so does U . Proof : Since + U has FI-J-supplement in M , then there exists a fully invariant X M , such that X + ( + U) = M , and X ( + U) X . Since is FI-J-supplemented module , then there exists Y such that (X + U ) (1) It is clear that every FI--J-supplemented is FI-J-supplemented . But the converse in general is not true , for example Z as Z-module . (2) as Z-module is FI--J-supplemented . , that is Y is FI--J-supplement of X + K in M. Next , we show that X + Y is FI--J-supplement of K in M. It is clear that M = K + (X +Y) , so it suffices to show that (X + Y) K X +Y. Since   (1))] we get herefore Y is FI--J-supplement of X in Thus is FI--J-supplemented module . Theorem To prove that M is FI--J-supplemented R-module, it is sufficient by the induction on n to prove this is the case when Thus suppose that Let X be any fully invariant submodule of M . Then so that has a -J-supplement 0 in M . Since is FI--J-supplemented , has a -J-supplement H in such that H is a direct summand of By lemma (3.23) , H is a -J-supplement of in M . Since is FI--J-supplemented, has a -J-supplement K in such that K is a direct summand of Again by lemma(3.23) , H + K is a -J-supplement of X in M . Since H is a direct summand of and K is a direct summand of it follows that H + K = H K is a direct summand of M . Thus is FI--J-supplemented .